In this work, we consider the stochastic Cauchy problem driven by the canonical $\alpha $-stable cylindrical Lévy process. This noise naturally generalises the cylindrical Brownian motion or space-time Gaussian white noise. We derive a sufficient and necessary condition for the existence of the weak and mild solution of the stochastic Cauchy problem and establish the temporal irregularity of the solution.
"Stable cylindrical Lévy processes and the stochastic Cauchy problem." Electron. Commun. Probab. 23 1 - 12, 2018. https://doi.org/10.1214/18-ECP134